I need a suggestion on how to detect how fast an object tethered to a rope is traveling when it's spun. Basically, take a tennis ball attached to a rope and spin the thing over your head. I need a way to have something inside of the tennis ball detect how fast it's spinning. Suggestions?

Is the length of the rope known and constant? Is the anchor point of the rope fixed? If everything works out in your favour, you could probably get a reasonable estimate of the speed by measuring the average acceleration over a few seconds. If things aren't ideal - the rope length isn't fixed and know, or the center isn't stationary, the next approach I can see is to measure the acceleration in three dimensions and calculate the net acceleration vector, and use a gyro to measure the angular velocity. The maths will take some working out, but in theory it would be possible to determine the angular velocity excluding rotation around the axis of the rope, and given the angular velocity and acceleration it is possible to calculate the radius and speed.

Length of rope is constant, however the center point isn't. This is exactly what I gave as an example, a ball tied to a rope being spun around. It's an idea for poi, for a performer to use. One in each hand, spinning around.

What they want is LEDs embedded in them that will change color based on the speed in which those things are being spun. There are several static displays on the market already, push a button to turn them on and spin away, but there's no interaction. If you want to change the display you end up having to stop, push the button again, and go on. So I was asked if I can make something that will change colors based on the speed at which they're being spun around.

In that case I'd have thought you could do that easily enough with just an accelerometer, don't bother trying to work out actual speed, just vary the output according to the recent average acceleration.

Have you thought of using this to produce a persistance-of-vision style display? That would be much harder to do, but I imagine the results could be quite impressive.

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From my limited experience of seeing people doing this sort of thing, it looks as if the way they pump the things up is very uneven and it would probably be possible to look for regular fluctuations in the acceleration and infer the phasing from that. This is likely to be a lot easier if you have an idea how long the rope is likely to be and can guess the approximate period from the average acceleration. (In other words, it may be possible to get PoV working without as much complexity as I'd thought, especially if your performers are willing to cooperate.)

I only provide help via the forum - please do not contact me for private consultancy.